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Experimental Demonstrator of the Uncertainty Principle Degree Project in Engineering Physics, First Level (SA104X)

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Experimental Demonstrator of the Uncertainty Principle Degree Project in Engineering Physics, First Level (SA104X) Experimental Demonstrator of the Uncertainty Principle Degree Project in Engineering Physics, First Level (SA104X) Philip Ekfeldt Anders Pettersson [email protected] [email protected] May 21, 2013 Supervisors: Marcin Swillo and Gunnar Bj¨ork Examinator: M˚artenOlsson Dept. of Applied Physics Royal Institute of Technology Abstract The goal of this project was to create an intuitive and clear demonstrator of the defining properties of quantum mechanics using single slit diffraction of light, which has quantum mechanical properties because of light's wave- particle duality. In this report we will describe the process and thoughts behind our project of creating a portable demonstration of the uncertainty principle. By designing and building both a physical setup with a laser, a slit, mirrors, lenses, beam-splitters, attenuators and cameras, and developing software which displays images from the cameras in a clear user interface with calculations we hope that students from high schools and gymnasiums that visit Vetenskapens Hus at Alba Nova will learn something new while using the demonstrator. CONTENTS Philip Ekfeldt, Anders Pettersson Contents 1 Introduction 2 1.1 Objective . 2 2 Theory 3 2.1 The Uncertainty Principle . 3 2.2 Theoretical Value of Uncertainty . 3 2.3 Uncertainty of Laser Source . 5 3 Method 6 3.1 Optical Setup . 6 3.1.1 Projecting Images to the Cameras . 8 3.2 Software . 10 3.2.1 Description of the User Environment . 10 3.2.2 Calculating the Standard Deviation of the Position . 11 3.2.3 Calculating the Standard Deviation of the Momentum 12 4 Results 14 5 Discussion 14 5.1 Physical Design . 14 5.2 Software Design . 15 5.3 Measured Results . 16 6 Conclusion 17 7 Acknowledgements 17 1 Philip Ekfeldt, Anders Pettersson 1 Introduction After James Maxwell formulated his equations that combined electricity and magnetism into a single theory at the end of the 19th century, the general consensus became that light travelled as an electromagnetic wave. It was not until Albert Einstein posted his theory on the photoelectric effect that the idea that light behaves both as a wave and as particles gained momen- tum again. Our demonstration is based on photon diffraction, the natural phenomena that allows photons to diffract in a single slit just like a wave to create a diffraction pattern. Combining this with the uncertainty principle of quan- tum mechanics, we can analyse our results and compare it to the inequality ¯h σxσpx ≥ 2 derived by Earle Hesse Kennard in 1927 [1]. The basis for this principle came from the groundbreaking work in quantum mechanics by German physicist Werner Heisenberg who earlier that same year gave the argument that such a limit for the deviation in position and momentum should exist. 1.1 Objective The primary goal of this project is to create a portable demonstrator of the ¯h uncertainty principle of quantum mechanics, σxσpx ≥ 2 , with the help of single slit diffraction of light. This demonstration is meant for showing high school students real quantum effects and educating the basic principles of quantum mechanics, which are visually clear in the diffraction pattern from the single slit experiment. To show these effects, the demonstration is divided into two parts. First we create the optical experiment, with a Helium-Neon laser, a slit, mirrors, lenses, attenuators and detectors. This setup shows the diffraction pattern on a white paper. Connected to the detectors we also have a computer with software capable of displaying the detected images and calculating the essential properties from these patterns which can then be related to the uncertainty principle. The tasks in this project were divided as follows. Philip was in charge of the optical experiment and writing the theoretical background, while Anders handled the programming of the software and the measurement analysis. It should be noted that we have worked together on this project and helped each other with our different tasks. 2 Philip Ekfeldt, Anders Pettersson 2 Theory 2.1 The Uncertainty Principle The uncertainty principle of quantum mechanics describes the inherent un- certainties in a particle's properties such as position and momentum due to particles' wave nature. What the principle states is that it is impossible to know or prepare precisely both a particle's position x and its momentum px along the same axis. There is a strict lower limit for the product of the ¯h uncertainties of these two properties: ∆x∆px ≥ 2 with ¯h being the reduced h −34 Planck constant 2π = 1:055 · 10 Js. In physics, this uncertainty ∆ is defined as the standard deviation σ, which is the notation we will use in this report [2]. This uncertainty relation can be seen in a single slit diffraction experiment. Sending light through a narrow slit, the light is diffracted and a diffraction pattern is seen on a screen on the other side. Making the slit less wide, thus reducing the uncertainty of the photons' position, makes the diffraction pattern wider indicating an increased uncertainty of their momentum in the direction of the slit width. We would expect this diffraction to occur due to the wave nature of light. The interesting thing is this: If you would send each photon by itself through the slit, it would yield the same result. The photons hitting the screen would create a diffraction pattern anyway if you would accumulate their impact positions. This shows that photons (and particles in general) have an inherent uncertainty in position and momentum. 2.2 Theoretical Value of Uncertainty The light intensity of the laser used in the project can be seen as having the form of a Gaussian function. If you open the slit completely so that none of the light is blocked, we can approximate the intensity function over the horizontal axis (x) as a Gaussian if we average the intensity value of the beam over the vertical axis (y). Suppose that this Gaussian intensity function (which is proportional to the probability distribution of the light) has a standard deviation of σx. From this follows that 2 − x I(x) / e 2σx2 : Since the complex amplitude of the wave A(x) is proportional to the square root of the intensity we get: 2 p − x A(x) / I(x) / e 4σx2 : Theory [3] shows that the far field intensity function can be found by apply- ing the Fourier transform to this complex amplitude function of the source, 3 2.2 Theoretical Value of Uncertainty Philip Ekfeldt, Anders Pettersson x0 with the frequency variable equalling ξ = λz , with z being the distance from the source to the measuring point on the z-axis and x0 being the distance from the z-axis at this point. x2 − F −4σ 2(πξ)2 2 −8σ 2(πξ)2 A(x) / e 4σx2 −! U(ξ) / e x −! I(ξ) / U(ξ) / e x ; where U(ξ) is the complex amplitude of the diffracted wave depending on ξ and I(ξ) is the related intensity which both also have the form of a Gaussian function. Comparing this to the original Gaussian function, we can see that ξ has a standard deviation of σ = 1 . ξ 4πσx A photon has the linear momentum p =hk; ¯ 2π with k being the wave vector λ . The momentum along the x-axis, px, for the diffracted beam can be calculated using 2π p = p sin θ = ¯h sin θ x λ with the average being zero, see Figure 1. Since x0 << z for the far field, x0 ≈ sin θ: z This means that ξ can be rewritten as sin θ ¯h ξ ≈ −! px = 2¯hξπ −! σpx = : λ 2σx As mentioned in section 2.1 we are only interested in the deviation of the momentum along the same axis as the deviation of the position, which is along the x-axis. Using the values for the standard deviations in the uncer- ¯h tainty inequality σxσpx ≥ 2 we find that this result represents the equality. This means that using a Gaussian beam one saturates the uncertainty in- equality. If we instead close the slit, the intensity function at the slit looks more like a rectangular function. This would return a diffraction pattern on the form of a sinc function squared, sin πwξ 2 I / sinc2(wξ) = πwξ by Fourier transforming, where w is the width of the slit. Calculating the standard deviation of the normalised I(ξ) one will find that the integral does not converge. This is because ξ is not limited and sin πwξ 2 ξ2 / sin2 πwξ πwξ 4 2.3 Uncertainty of Laser Source Philip Ekfeldt, Anders Pettersson Figure 1: Single slit wave diffraction in the integral does not converge when ξ goes toward infinity. What you need to do is look at the angular distribution of the intensity (since θ is π π limited between − 2 ≤ θ ≤ 2 ), which you can get by integrating the complex amplitude of the light over the slit width. 2.3 Uncertainty of Laser Source In an ideal case the beams from a light source would be parallel, but in real- ity, due to the uncertainty principle, any finite source has a divergence. The specification of the laser has two interesting parameters for the uncertainty; 1 1 beam diameter ( e2 ) of 0.65 mm and beam divergence ( e2 ) of 1.24 mrad. The beam radius (0.325 mm) r0 is half of the beam diameter and the beam divergence in one direction (0.62 mrad) from the z-axis is half of the stated 1 beam divergence. The description ( e2 ) means that where the absolute value of x (or r in the two dimensional case) equals the beam radius, the intensity of the beam is e−2 times the maximum value which occurs at the center of the beam.
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